Fibonacci Spiral: The Sequence, the Golden Ratio & the Squares That Draw It

The rule: each number is the sum of the two before it

Named after Leonardo of Pisa — nicknamed Fibonacci — who introduced it to Europe in 1202, the sequence follows one deceptively simple rule: F(n) = F(n−1) + F(n−2), starting from F(1) = 1 and F(2) = 1. Add the last two terms to get the next one, and you generate 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and onward forever. Fibonacci originally described the sequence through a population of idealised rabbits in his book Liber Abaci — every month, each mature breeding pair produces a new pair, and the total pair-count each month is exactly this sequence.

The golden ratio hides inside the sequence

Divide each Fibonacci number by the one before it, and the further along you go, the closer that ratio creeps toward a single fixed value — the golden ratio, φ (phi):

φ = (1 + √5) / 2 ≈ 1.6180339887…
89/55  ≈ 1.6182
144/89 ≈ 1.6180
233/144 ≈ 1.61806  (already accurate to 4 decimal places)

The ratio never lands exactly on φ, only ever converges toward it — but the convergence is extremely fast. This isn't a numerical accident: it falls straight out of solving the recurrence relation algebraically, and φ is the unique positive number satisfying φ² = φ + 1, meaning φ is literally one plus its own reciprocal.

Stacking squares: how the spiral gets drawn

The classic Fibonacci spiral is built geometrically, one square at a time, using the sequence itself as side-lengths. Draw a 1×1 square, then another 1×1 beside it. Together they span a 2-unit edge, so the next square is 2×2. Its edge plus the previous gives 3, so the next square is 3×3 — then 5×5, 8×8, 13×13, each one exactly the sum of the two squares before it, spiralling outward around a common corner. Inscribe a quarter-circle arc inside each square, corner to corner, and the arcs join seamlessly into one continuous curve: the Fibonacci spiral.

The "most irrational" number and why it matters

The golden ratio is also the limit of the simplest possible continued fraction, [1; 1, 1, 1, 1, …], and its successive convergents are exactly the ratios of consecutive Fibonacci numbers — 1/1, 2/1, 3/2, 5/3, 8/5, 13/8… This makes φ provably the real number that is hardest to approximate well with any simple fraction, since its continued-fraction expansion uses only 1s, the slowest-converging pattern possible. That single mathematical property — not aesthetics — is why the golden ratio and the Fibonacci sequence that approaches it turn up wherever a system needs to avoid clean numerical coincidences, from plant growth angles to spiral shell geometry.

Frequently asked questions

Try it live

Everything above runs in your browser — open Fibonacci Spiral and watch the sequence, the golden ratio and the stacked squares come together in one interactive drawing. Nothing is installed, nothing is uploaded.